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Under simple uniform hashing, load factor summarizes average occupancy and determines expected search work for separate chaining.
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Formula
n keys; m buckets; α and expected key comparisons are dimensionless counts
| Symbol | Meaning | Unit |
|---|---|---|
| Number of stored keys | count | |
| Number of table buckets | count | |
| Average keys per bucket, n/m | dimensionless | |
| Expected search work under chaining | key comparisons |
\alpha=\frac{n}{m},\quad E[T_{\mathrm{suche}}]=\Theta(1+\alpha)With chained buckets and simple uniform hashing, the expected chain length is α=n/m and search takes Θ(1+α) expected work. An unsuccessful search inspects about α chain elements; successful-search cost also depends on insertion and query distributions. Constant expected time follows only when α stays bounded. This is an expectation under a hashing assumption, not a guarantee for every input or adversarial hash function.
With chained buckets and simple uniform hashing, the expected chain length is α=n/m and search takes Θ(1+α) expected work. An unsuccessful search inspects about α chain elements; successful-search cost also depends on insertion and query distributions. Constant expected time follows only when α stays bounded. This is an expectation under a hashing assumption, not a guarantee for every input or adversarial hash function.
Under simple uniform hashing, load factor summarizes average occupancy and determines expected search work for separate chaining.
Defining relation
The equation states the defining relation within its stated domain.
Use the example for Hash table: load factor and expected search time to explain which inputs determine the requested output.
The example connects concrete inputs to its result.
Under simple uniform hashing, load factor summarizes average occupancy and determines expected search work for separate chaining.
The relation applies only under its stated assumptions.
Applying the relation to a different case without checking its stated assumptions.
Check the validity range before substituting values: With chained buckets and simple uniform hashing, the expected chain length is α=n/m and search takes Θ(1+α) expected work. An unsuccessful search inspects about α chain elements; successful-search cost also.
Formula cluster
Under simple uniform hashing, load factor summarizes average occupancy and determines expected search work for separate chaining.
For n=120 keys and m=30 buckets, α=120/30=4. Under the uniform-hashing model, the mean chain length is four and the search order is Θ(5), namely Θ(1+4). A particular chain can be much shorter or longer.
Estimate capacity growth and expected collision cost for chained hash tables.
Here is how to work through a typical Hash table: load factor and expected search time (α = n/m; erwartete Suche Θ(1+α)) task step by step:
Task
Use the example for Hash table: load factor and expected search time to explain which inputs determine the requested output.
Solution path
The example connects concrete inputs to its result.
Under simple uniform hashing, load factor summarizes average occupancy and determines expected search work for separate chaining.
The relation applies only under its stated assumptions.
The relation \alpha=\frac{n}{m},\quad E[T_{\mathrm{suche}}]=\Theta(1+\alpha) summarizes Under simple uniform hashing, load factor summarizes average occupancy and determines expected search work for separate chaining. Read it as a model for defined quantities, not as an automatic claim about every real situation. In the example For n=120 keys and m=30 buckets, α=120/30=4. Under the uniform-hashing model, the mean chain length is four and the search order is Θ(5), namely Θ(1+4). A particular chain can be much shorter or longer., symbols are connected to concrete values and the result is interpreted in context. Validity limit: With chained buckets and simple uniform hashing, the expected chain length is α=n/m and search takes Θ(1+α) expected work. An unsuccessful search inspects about α chain elements; successful-search cost also Source: MIT OpenCourseWare 6.006, Lecture 8: Hashing I, simple uniform hashing and load factor.
First write down the requested quantity and all given values with their symbols. Substitute them into the defining relation and check unit consistency. The example For n=120 keys and m=30 buckets, α=120/30=4. Under the uniform-hashing model, the mean chain length is four and the search order is Θ(5), namely Θ(1+4). A particular chain can be much shorter or longer. provides a worked check: formula and result must have compatible scale and meaning; round only at the end. Validity limit: With chained buckets and simple uniform hashing, the expected chain length is α=n/m and search takes Θ(1+α) expected work. An unsuccessful search inspects about α chain elements; successful-search cost also Source: MIT OpenCourseWare 6.006, Lecture 8: Hashing I, simple uniform hashing and load factor.
The formula applies only when quantities are defined as in the source and background. Check boundary cases, measurement conditions and whether an approximation rather than equality is intended. Substitution can be algebraically correct yet technically wrong when assumptions fail or the population or system boundary changes. Validity limit: With chained buckets and simple uniform hashing, the expected chain length is α=n/m and search takes Θ(1+α) expected work. An unsuccessful search inspects about α chain elements; successful-search cost also Source: MIT OpenCourseWare 6.006, Lecture 8: Hashing I, simple uniform hashing and load factor.
The stated units are: n keys; m buckets; α and expected key comparisons are dimensionless counts. Convert values to a consistent unit system before substitution; counts, percentages and dimensionless ratios are not interchangeable. The variable list states the role of each symbol. Dimensional analysis catches many errors but does not replace checking the formula's assumptions. Validity limit: With chained buckets and simple uniform hashing, the expected chain length is α=n/m and search takes Θ(1+α) expected work. An unsuccessful search inspects about α chain elements; successful-search cost also Source: MIT OpenCourseWare 6.006, Lecture 8: Hashing I, simple uniform hashing and load factor.
Comparison helps when both quantities address the same question but emphasize different properties. Report the definition, input data, threshold or averaging with the value. The source MIT OpenCourseWare 6.006, Lecture 8: Hashing I, simple uniform hashing and load factor. supports the definition used here; other conventions can differ. Choose the measure to match the learning or analysis goal. Validity limit: With chained buckets and simple uniform hashing, the expected chain length is α=n/m and search takes Θ(1+α) expected work. An unsuccessful search inspects about α chain elements; successful-search cost also Source: MIT OpenCourseWare 6.006, Lecture 8: Hashing I, simple uniform hashing and load factor.
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